This invention relates generally to medical imaging systems and methods and more particularly to system and methods for filtering a measurement of density of an object.
Typically, in computed tomography (CT) imaging systems, a gantry includes an X-ray source that emits a fan-shaped beam toward an object, such as a patient. The beam, after being attenuated by the patient, impinges upon an array of radiation detectors. The intensity of the attenuated beam radiation received at the detector array is typically dependent upon the attenuation of the X-ray beam by the patient. Each detector element of the detector array produces a separate electrical signal indicative of the attenuated beam received by each detector element. The electrical signals are transmitted to a data processing unit for analysis which ultimately results in a formation of an image.
Generally, the X-ray source and the detector array are rotated with a gantry within an imaging plane and around the patient. X-ray sources typically include X-ray tubes, which conduct a tube current and emit the X-ray beam at a focal point. X-ray detectors typically include a collimator for collimating X-ray beams received at the detectors, a scintillator for converting X-rays to light energy adjacent the collimator, and photodiodes for receiving the light energy from the adjacent scintillator. The photodiodes convert the light energy into electrical energy that represents projection data. A projection datum at a projection angle θ and a projection displacement t is represented as p(θ,t). A data acquisition system (DAS) receives the projection data from the detectors. One of the CT imaging systems is used to reconstruct a cross-section of the patient object by measuring the projections through the patient.
It is generally difficult to measure the projection datum p(θ,t) directly. Instead, the projection data is obtained by counting a number of photons impinging on the detectors. A measurement λ(θ,t) at the projection angle θ and the projection displacement t is proportional to a number of the photons. The measurement λ(θ,t) is corrupted by various kinds of noises, such as, a photon counting noise often modeled by a Poisson distribution or an additive noise often modeled by a Gaussian distribution. The additive noise is generated from dark current in the DAS, interference noise in cables of the CT imaging systems, and other sources. If an image is reconstructed by applying the measurement λ(θ,t) with the noises, shading artifacts are visible in the image.
Usually an expected value of λ(θ,t) is related to the projection datum by an equation λ(θ,t)=E[λ(θ,t)]=exp(−p(θ,t))λT  (A)
where E denotes an expectation, λ(θ,t) denotes the expected value of λ(θ,t), and λT denotes a number of photons that are impinging on the patient from the x-ray source. In practice, a value of λT may vary with the projection datum p(θ,t). Alternatively, in theory, λT may be presumed to be a constant. The projection datum p(θ,t) may be computed from the expected value of λ(θ,t) by using an equation
                              p          ⁡                      (                          θ              ,              t                        )                          =                  -                      log            ⁡                          (                                                                    λ                    _                                    ⁡                                      (                                          θ                      ,                      t                                        )                                                                    λ                  r                                            )                                                          (        B        )            
where log represents a logarithmic (log10) function. A difficulty in applying equation (B) is that a value of λ(θ,t) is not known because a noisy value of λ(θ,t) is directly measured.
A first approach used to determined the projection datum p(θ, t) is to use an approximation λ(θ,t)≅λ(θ,t). If λ(θ,t) is large, the approximation is useful because a value of a signal within the electrical energy is large compared to the noises. However, when the patient being imaged is dense and/or large, the attenuation may be large, and the value of λ(θ,t) is small. When λ(θ,t) is small, λ(θ,t) may have a large percentage error and/or may be negative. The measurement λ(θ,t) may have a large percentage error and/or may be negative because the additive noise in one of the CT imaging systems becomes significant as compared to the photon counting noise. When λ(θ,t) is negative, the approximation of cannot be used because the logarithm of a negative number cannot be computed in equation B.
A second approach is to obtain a plurality of λ(θ,t)s and average the λ(θ,t)s. The noises in one of the λ(θ,t)s are generally independent of the noises in any of the remaining of the λ(θ,t)s and so by averaging the λ(θ,t)s, the expected value λ(θ,t) is obtained. However, the second approach has a number of disadvantages. A first one of the disadvantages is that the second approach uses a higher amount of time to obtain the λ(θ,t)s than a time used to obtain the measurement λ(θ,t) because λ(θ,t) is measured multiple times. A second of the disadvantages is that the second approach exposes the patient to harmful radiation when the λ(θ,t) is obtained multiple times. The harmful radiation may result in greater health risk to the patient and the health risk is undesirable.
A third approach is to smooth λ(θ,t) with neighboring values, such as λ(θ+1,t) or λ(θ,t+1), of λ(θ,t). The measurement λ(θ,t) and the neighboring values have nearly equal angles. The measurement λ(θ,t) and the neighboring values also have nearly equal displacements. A smoothing filter is applied to λ(θ,t) and the neighboring values to average λ(θ,t) and the neighboring values. The smoothing filter is applied to produce a smoothed version of the measurement λ(θ,t) and the smoothed version is denoted by {circumflex over (λ)}(θ,t). The projection datum p(θ,t) may be obtained by using an approximation λ(θ,t)≅{circumflex over (λ)}(θ,t). The smoothing filter can reduce the noises by averaging but it has flaws. A first one of the flaws is that the smoothing filter may not provide the smoothed version that is positive in value because a group of negative measurements may fall near each other. A second one of the flaws is that averaging λ(θ,t) with the neighboring values tends to blur a resolution of a reconstruction in regions, having high signal-to-noise measurements, that do not require averaging. A third one of the flaws is that a local mean value of a local region may not be preserved, which could lead to shading artifacts in the image.
A Floyd-Steinberg algorithm preserves the local mean value during filtering of the noises. The Floyd-Steinberg algorithm preserves the local mean value by applying binarization. A direct binary search algorithm also preserves the local mean value during filtering of the noises. However, the Floyd-Steinberg error diffusion algorithm and the direct binary search algorithm are generally used for digital halftoning in printing.